Let us start by explaining Bellissard's
approach in a nutshell. The magnetic (non-random) \(C^*\) -algebra considered
in [1]},
the so called noncommutative Brillouin zone, is the subalgebra
\(\widetilde{{C}}_B\subset {B}\big (\ell ^2\big (\mathbb {Z}^2\big )\big )\)  generated by the discrete
magnetic translations.The name magnetic translations is common in
the condensed matter community since the works of Zak [2]}, [3]}. Mathematically, they are known as Weyl systems. The
subscript \(B\)  in the symbol \(\widetilde{{C}}_B\)  refers to the
strength of a constant magnetic field perpendicular to the plane. The \(C^*\) -algebra
\(\widetilde{{C}}_B\)  is unital and is endowed with a natural finite
trace \({T}_B\)  known as the trace per unit volume (see [4]} for
a recent review). The NGC built in [1]} is
based on the spectral triple
\(\big (\widetilde{\rho },\widetilde{{H}}_2,\widetilde{D}\big )\)  where
\(\widetilde{{H}}_2:=\ell ^2\big (\mathbb {Z}^2\big )\otimes 2\)  is a separable Hilbert space,
\(\widetilde{\rho }\)  is the diagonal representation of \(\widetilde{{C}}_B\)  on
\(\widetilde{{H}}_2\)  defined by \(\widetilde{\rho }(A):=A\otimes {\bf 1}_2\)  and
\(\widetilde{D}\)  is the Dirac operator defined by
\(\widetilde{D} := x_1 \otimes \sigma _1 + x_2 \otimes \sigma _2 = \left(\begin{matrix}0 & x_1-\,\mathrm {i}\,x_2 \\x_1+\,\mathrm {i}\,x_2 & 0\end{matrix}\right),\) 
